Planar relative Schottky sets and quasisymmetric maps (Q2881018)

The author studies the geometry of maps within an interesting class of fractal sets, called relative Schottky sets. Given a domain \(\Omega\subset \mathbb S^n\) and a subset \(S\subseteq \Omega\), one says that \(S\) is a relative Schottky set in \(\Omega\) if \(\Omega\setminus S\) is a union of open balls with disjoint closures. This paper focuses on the planar case \(n=2\) and on the relative Schottky sets of zero area.NEWLINENEWLINEThe restriction of the spherical metric endows \(S\) with the structure of a metric space. A homeomorphism \(f: S\to S'\) between two metric spaces is locally quasisymmetric if its restriction to every compact subset is quasisymmetric; the latter means that \(f\) distorts relative distances in a controlled manner. The main results of this paper concern the smoothness and rigidity of locally quasisymmetric maps between relative Schottky sets. Two of them are quoted below. NEWLINENEWLINENEWLINENEWLINE Theorem 1.2. Suppose that \( S\) is a relative Schottky set of measure zero in a Jordan domain \(\Omega\subseteq \mathbb C\). Let \(f : S\to S'\) be a locally quasisymmetric orientation-preserving map from \(S\) to a relative Schottky set \(\widetilde S\) in a Jordan domain \(\widetilde{ \Omega}\subseteq \mathbb C\). Then \(f\) is conformal in \(S\) in the sense that for every \(p\in S\), the limit NEWLINE\[NEWLINEf'(p):=\lim_{q\to p, q\in S}\frac{f(q)-f(p)}{q-p}NEWLINE\]NEWLINE exists and is nonzero. Moreover, \(f\) is locally bi-Lipschitz in \(S\) and \(f'\) is continuous in \(S\).NEWLINENEWLINEThe author conjectures that the assumption of \(\Omega\), \(\widetilde{\Omega}\) being Jordan domains can be removed, and that \(f\) is in fact infinitely differentiable. NEWLINENEWLINENEWLINENEWLINE Theorem 1.7. Suppose that \(S\) and \(\widetilde S\) are relative Schottky sets of measure zero in the unit disc \(\mathbb U\). Then every locally quasisymmetric orientation-preserving map between \(S\) and \(\widetilde S\) is the restriction of a Möbius transformation.